Delta function potentials

I have a potential which is zero everywhere except at -2a , -a , 0 , a , 2a on the x-axis where there is an attractive delta potential at each of the 5 points. I know there is a maximum of 5 bound states. I know there can be no nodes for |x| > 2a and a maximum of one node between each delta potential. My question is ; why cant there be one further excited state - an antisymmetric state with a node at x=0 and 4 further nodes ; one between each delta potential ?

Staff: Mentor

Antisymmetric wavefunction would have the form of sine so 1st derivative would be cos(0) =1 , 2nd derivative would be sin(0)=0. I can't see how that helps and the same would happen for the 1st and 3rd excited states as well

That state will have positive energy. The wave function changes "too fast" between the node at x=0 and the nodes between 0 and +/-a to have negative energy. It is also not an energy eigenstate; it will have nonzero inner product with both positive and negative energy states.

Antisymmetric wavefunction would have the form of sine so 1st derivative would be cos(0) =1 , 2nd derivative would be sin(0)=0. I can't see how that helps and the same would happen for the 1st and 3rd excited states as well

As Avodyne says, a wave function that looks like the sine function has positive energy so will not be bound. A bound state wave function must look like a linear combination of exponentials in regions where V=0.